Coloring and IGP

One of the first IGP activities I did with kids at school was coloring IGP. (I also did the same activity with teachers at Math Circle in Athens, Ohio.) The activity is simple, but it can progress in levels of difficulty.

When I do this activity, I usually give a plain IGP to kids and ask them to color it, with the condition that no two motifs or shapes that are adjunct should have same color. The general form of this question is familiar to mathematicians and we call it the Four-Color Theorem (as you can see from the name we only need four colors to color any map without any two regions that share borders having the same color). In the case of IGP it is simpler. Based on my experience, most IGP can be colored by using only two colors and this hypothesis was verified after I read about it in a famous book by Grunwald and Shepard called Tilings and Patterns, which people call the bible of patterns.

Here is what I do when I give kids these activities:

First Activity:

Give kids a sheet of paper that has the IGP on it.

Ask them to color it in a way that no two shapes share a link or curve that has the same number.

What number do they get? Did everyone in the class get the same number? And why?

Second Activity:

This is similar to the first one but this time you will give a more complex pattern, and they will need at least three colors. Ask them why we cannot use only two colors?

Third Activity:

Give them a pattern that consists of more than two different motifs. You will ask them to color the pattern with the same conditions as before, but this time you will require that one of the motifs should always have the same color. This requirement will lead to further investigation.

Fourth Activity:

One way to encourage kids to think more abstractly about coloring IGP is to change the question in a way that kids will use a blank sheet of paper try to create a graph related to the IGP in this way:

For each shape the child will draw a circle and if two shapes share a line then the child should draw a line between the two circles that represents the shapes.

This way the question will be changed from coloring the pattern to only coloring the circle. Also the number in this case will be called a “chromatic number”. Here is an amazing activity that has been done with seven-year-olds by Oxford math professor Joel David Hamkins

You can download some IGP and print it so kids play with coloring it. The link can be found here.